Minimum-Phase Polynomials

A filter is minimum phase if both the numerator and denominator of its
transfer function are
minimum-phase polynomials
in :

The case is excluded because the polynomial cannot be minimum
phase in that case, because then it would have a zero at
unless all its coefficients were zero.

As usual, definitions for filters generalize to definitions
for signals by simply treating the signal as an impulse
response:

Note that every stable all-pole filter
is
minimum phase, because stability implies that is minimum
phase, and there are ``no zeros'' (all are at ).
Thus, minimum phase is the only phase available to a stable all-pole
filter.

The contribution of minimum-phase zeros to the complex cepstrum
was described in §8.8.